Introduction

Isaki¹ was the first to discuss the issue of estimates of population variance when auxiliary variable is present and proposed the classical ratio method of estimation for population variance which utilize the ratio population and sample variance of auxiliary variable as auxiliary information which is strongly correlated with the study variable. Numerous authors have modified ratio estimators for finite population variance when an auxiliary variable is present in the literature, authors like Evans³, Hansen and Hurwitz⁵, Prasad and Singh¹⁴. Prasad and Singh¹⁴ modified the work of Isaki⁶ by incorporating coefficient of kurtosis into his work. Ever since then several authors have been making efforts to develop estimators with greater efficiency using auxiliary information for improvement and modification, other authors include Kadilar and Cingi⁷, Gupta and Shabbir⁴, Tailor and Sharma¹⁶, Khan and Shabbir⁸, Yadav et al.¹⁷, Subramani¹⁵, Audu et al.¹, Maqbool and Shakeel⁹, Muili et al.^12,11, Bhat et al.², and Muili et al.¹⁰. Muili et al.¹⁰ modified the work of Bhat et al.² by incorporating First quartile, Downton’s method, Deciles, Mid-range and Percentiles into the developed estimators for the purposed of precision.

To provide estimates that are closer to the population mean of the study variable, we proposed arithmetic mean estimators from a family of ratio type estimators of finite population variance in this study.

Consider a finite population G={G₁, G₂,…, G_N}, a set of N units, where G₁ = {X_i,Y_i}, I = 1,2,…,N , has a pair values. Y is the study variable and is associated with the auxiliary variable X, where y_i = {y₁,y₂,…, y_n} and x_i = {x₁,x₂,……,x_n}  are the n sample values. The sample means for the study and the auxiliary variables are ?  and x, respectively. S_x²  and S_y² are the population mean squares of X and Y respectively and s_x² and s_y² represent the sample mean square for the size n randomly selected, without replacement. Y: Study variable, n: sample size, N: population size, X: Auxiliary variable, x, ? : Auxiliary and study variables sample means, x, ?: Auxiliary and study variables population means, ƒ : Sampling fraction, p : Correlation coefficient C_y, C_x : Coefficient of variation of study and auxiliary variable, β_1(x) : Skewness, Q₁ : First quartile of auxiliary variable, Q₂ : Second quartile, Q₃ :Third quartile, QD : Quartile deviation, β_2(x) : Kurtosis, TM: Tri-mean, M_d : Median of auxiliary variable, Mx : Maximum value of auxiliary variable, Q₁ : Hodges lehman estimator, MR: Population mid-range of auxiliary variable, Q₁: Downton’s method, P_i: Percentile of auxiliary variable.



Literature review

The population's variance's unbiased estimator is

The estimator's t₀ bias and mean square error (Variance) are provided by

When the population and sample variance of the auxiliary variable are available, Isaki⁶'s proposed ratio estimator for estimating population variance of the study variable is defined as

The estimator’s t_R bias and mean square error are

Kadilar and Cingi⁷ developed class of ratio estimators taking into account the coefficient of variation and kurtosis of the auxiliary variable as auxiliary information for improvement in the estimation of population variance and it is defined as

The estimator’s t _KCi bias and mean square error are:

Subramani¹⁵ developed a generalized modified ratio estimator that uses the median and coefficient of variation of the auxiliary variable to estimate the finite population variance of the study variable in order to enhance the precision of the estimate as.


The estimator’s t_s bias and mean square error are:

Bhat et al.² modified robust estimators for the estimation of population variance utilizing a linear combination of Downton's approach and deciles of the auxiliary variable in order to improve the precision of the estimate as well as the estimation of the study's finite population variance.

The estimator’s t_Bi bias and mean square error are:



Where, b₁ = (D + D₁),     b₂ = (D + D₂),   b₃ = (D + D₃),   b₄ = (D + D₄),   b₅ = (D + D₅),   b₆ = (D + D₆),   b₇ = (D + D₇), b₈ = (D + D₈), b₉ = (D + D₉), b₁₀ = (D + D₁₀),

Muili et al.¹⁰ developed a family of ratio type estimators for finite population variance estimation utilizing the parameters of the auxiliary variable, first quartile, downton’s method, deciles, mid-range and percentiles as

The estimator’s t_Ji  bias and mean square error are:

Where, h₁ = (D₁₀ + P₁),   h₂ = (D₁₀ + P₅),  h₃ = (D₁₀ + P₁₀),  h₄ = (D₁₀ + P₁₅),  h₅ = (D₁₀ + P₂₀),  h₆ = (D₁₀ + P₂₅),  h₇ = (D₁₀ + P₃₀),  h₈ = (D₁₀ + P₃₅),  h₉ = (D₁₀ + P₄₀),  h₁₀ = (D₁₀ + P₄₅),  h₁₁ = (D₁₀ + P₅₀),  h₁₂ = (D₁₀ + P₅₅),  h₁₃ = (D₁₀ + P₆₀),  h₁₄ = (D₁₀ + P₆₅),  h₁₅ = (D₁₀ + P₇₀),  h₁₆ = (D₁₀ + P₇₅),  h₁₇ = (D₁₀ + P₈₀),  h₁₈ = (D₁₀ + P₈₅),  h₁₉ = (D₁₀ + P₉₀),  h₂₀ = (D₁₀ + P₉₅),

Proposed Estimators

We proposed arithmetic mean estimators of family of ratio type estimators of finite population variance when correlation between study and auxiliary variable is either positive or negative after studying and being inspired by Muili et al.¹⁰'s work.

The suggested estimators can be expressed generally as follows:

Properties (Bias and MSE) of the proposed estimators

To derive the bias and MSE of the proposed estimators we use the following definitions e’s sampling errors as

Expressing (40) in terms of sampling errors, we have

By simplifying (43) up to first order approximation, we have

By expanding, simplifying and subtracting S_y² from both sides of (44), we have

By taking the expectations from both sides of (45), one may get the estimator's t_Ai bias as

The MSE of t_Ai is obtained by squaring both sides of (45) and taking expectation.

By solving for z_i and differentiating (48) with respect to z_i equating to zero, we obtain

where,

By substituting (49) into (48), getting the estimator's minimum MSE t_Ai as

Efficiency comparisons

The proposed estimators t_Ai are more efficient than the existing estimators, if the following conditions are satisfied

Empirical study

To evaluate the effectiveness of the proposed estimators over existing ones, empirical study is carried out using a real life data set as

Data : [Source: {Murthy¹³ , P.228]

X: Fixed capital

Y: Output of Eighty factories

Table 1 MSEs and PREs of Existing and Proposed Estimators

Results and Discussion

Using the aforementioned real-life dataset, Table 1 displays the MSEs and PREs of the proposed and existing estimators. The findings showed that, in comparison to the existing estimators, the proposed ones have lower MSEs and greater PREs. This suggests that the proposed estimators have a better chance of generating estimates that are more in line with the study variable's actual population mean.

Conclusion

In this study, we proposed arithmetic mean estimators of a family of ratio type estimators of finite population variance, the properties (Biases and MSEs) of the proposed estimators were derived and from the results of empirical study, it was obtained that the proposed estimators are more efficient than sample mean estimator, ratio estimator, Kadilar and Cingi⁷ estimator, Subramani¹⁵ estimator, Bhat et al.² and Muili et al.¹⁰ estimators. Hence, the proposed estimators are recommended for use in real life situation.

Conflict of Interest

The authors declare no conflict of interest

Funding Sources

No financial supported was received by the authors for the research, authorship and publication of this article

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